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The 2010 IEEE/RSJ International Conference on

Intelligent Robots and Systems

October 18-22, 2010, Taipei, Taiwan

Abstract—Concentric tube robots are a subset of continuum

robots constructed by combining pre-curved elastic tubes. As

the tubes are rotated and translated with respect to each other,

their curvatures interact elastically, enabling control of the

robot’s tip configuration as well as the curvature along its

length. This technology is projected to be useful in many types

of minimally invasive medical procedures. Because these robots

are flexible by design, they deflect considerably when applying

forces to the external environment. Thus, in contrast to rigid-

link robots, their kinematic and static force models are

coupled. This paper derives a multi-tube quasistatic model that

relates tube rotations and translations together with externally

applied loads to robot shape and tip configuration. The model

can be applied in robot design, procedure planning as well as

control. For validation, the multi-tube model is compared

experimentally to a computationally-efficient single-tube

approximate model.

I. INTRODUCTION

HE goal of minimally invasive surgery (MIS) is to

interact with tissue deep inside the body while

minimizing collateral damage to surrounding tissues. In

contrast to open surgery in which access is gained by

making large incisions, MIS involves entering the body

through small incisions and, whenever possible, following

natural passages through the tissues to reach the surgical

site. Manual and robotic catheters are successful examples

of an MIS instrument technology which have been

specifically developed for procedures inside the vasculature

[1],[2].

There are many medical procedures that could benefit

from an instrument technology with the ability of catheters

to follow complex curves, but which require much more tip

stiffness than that of a catheter. These include structural

repairs inside the heart and tissue removal inside the brain.

Few instrument technologies exist, however, that possess

significant tip stiffness in combination with the ability to

assume 3D curves inside the body. Conventional surgical

Manuscript received February 28, 2010. This work was supported by the

National Institutes of Health under grants R01HL073647 and

R01HL087797.

J. Lock is with Biomedical Engineering, Boston University, Boston, MA

02215 USA (lockj@bu.edu).

G. Laing is with Mechanical Engineering, Boston University, Boston,

MA 02215 USA (glaing@bu.edu).

M. Mahvash is with Harvard Medical School, Boston MA 02115 USA

(mohsenmahvash@gmail.com).

P. Dupont is with Cardiovascular Surgery, Children’s Hospital Boston,

Harvard Medical School, Boston MA 02115 USA

(Pierre.Dupont@childrens.harvard.edu).

robots, for example, possess high stiffness, but consist of

straight shafts comparable to traditional laparoscopic tools

[3]. To address this shortcoming, bending snake-like robotic

extensions have been proposed and constructed for

mounting at the tip of the straight shaft [4]. A novel,

alternate approach consists of a robotic sheath that can be

extended along a 3D curve [5].

Concentric tube robots offer a good compromise between

shape control and stiffness. As illustrated by the example of

Fig. 1, they can be constructed to possess a full six degrees

of freedom at their tip while also enabling control of

curvature along their length. Furthermore, they can be

constructed with diameters comparable to catheters and

lengths sufficient to reach anywhere inside the body while

achieving a tip stiffness several orders of magnitude greater

than that of a catheter. The lumen of the innermost tube can

house additional tubes and wires for controlling articulated

tip-mounted tools.

Fig. 1. Concentric tube robot comprised of four telescoping sections that

can be rotated and translated with respect to each other.

Concentric tube robots, like steerable catheters [1],[2] and

snake-like multi-backbone devices [4], are continuum

robots. In comparison to traditional robot arms, this class of

robots lacks distinct links and joints. Continuum robots

possess the shape of a smooth curve whose curvature can be

controlled by adjusting the internal deformation of

mechanically coupled elastic components of the body.

Consequently, the kinematic modeling of continuum

robots cannot be formulated solely in terms of constrained

motion between rigid bodies, but must also incorporate

deformation modeling of the elastic components [1],[4],[6]-

[9]. For concentric tube robots, the deformation is that of the

individual tubes [6]-[9].

Owing to both the complexity of the modeling problem as

well as to the desire to derive numerically efficient models

for real-time control, a succession of kinematic models of

increasing complexity have been proposed over the last few

years as described in [8]-[9]. While providing significantly

improved accuracy over earlier models, these new models

are considerably more complex. They consist of second-

Quasistatic Modeling of Concentric Tube Robots

with External Loads

Jesse Lock, Genevieve Laing, Mohsen Mahvash and Pierre E. Dupont, Senior Member, IEEE

T

978-1-4244-6676-4/10/$25.00 ©2010 IEEE2325

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order differential equations with split boundary conditions.

To achieve computational efficiency, these equations can be

pre-computed over the workspace and stored either in the

form of a functional approximation or as a lookup table [8].

The inverse kinematic problem can be solved efficiently by

root finding on the approximate forward solution [8].

Alternately, an inverse functional approximation or lookup

table can be similarly constructed.

While the kinematic models of [8]-[9] assume that there

is no external loading applied to the robot (see [10] for an

exception), applications in minimally invasive surgery can

be expected to involve loads applied along the robot’s length

as well as at its tip. Unlike robots whose links can be

approximated as rigid bodies, however, the kinematic and

static force models of continuum robots cannot be

decoupled.

Thus, when considering the important case of external

loads applied to the robot, the model for implementing

position, force or impedance control takes the form of a

coupled 3D beam-bending problem in which the kinematic

input variables (tube rotations and displacements at the

proximal end) enter the problem as a subset of the boundary

conditions. The remaining boundary conditions are

comprised of point forces and torques applied to the distal

ends of the tubes. Contact along the robot’s length (e.g.,

with tissue) generates additional distributed and point loads.

In contrast to the models of [8]-[9], the inclusion of

external loading significantly increases the number of state

variables that must be integrated along the lengths of the

tubes. As an alternative to this full-order model, a

computationally-efficient approximate model that can be

applied to all types of continuum robots has been proposed

and successfully implemented for concentric-tube robot

stiffness control [11]-[13]. In this approach, the continuum

robot is modeled as a single Cosserat rod with properties

along its length corresponding to the composite stiffnesses

and initial curvatures of the unloaded robot.

The contributions of this paper are the derivation of a

multi-tube quasistatic model as well as a computational and

experimental comparison of the multi-tube model with the

single-tube model of [11]-[13]. The paper is arranged as

follows. Section II derives the multi-tube externally-loaded

model. Section III presents the simplified single-tube

approximate model. Section IV provides an experimental

comparison of the models. Conclusions are presented in

Section V.

II. QUASISTATIC MULTI-TUBE MODEL

The multi-tube model derived here can be interpreted as

an extension of the unloaded model presented in [8]. It

includes bending and torsion for an arbitrary number of

tubes whose curvature and stiffness can vary with arc length.

Effects that are neglected include shear of the cross section,

axial elongation, nonlinear constitutive behavior and friction

between the tubes. Note that these effects are neglected, but

are not necessarily all negligible.

In the remainder of the paper, subscript indices

1,2,...,

in are used to refer to individual tubes with tube 1

being outermost and tube n being innermost. Arc length, s,

is measured such that s = 0 at the proximal end of the tubes.

The total length of each tube is designated by Li.

As illustrated in Fig. 2, for two tubes, material coordinate

frames for each cross section can be defined as a function of

arc length s along tube i by defining a single frame at the

proximal end,

(0)

, such that its z axis is tangent to the

tube’s centerline. Under the unrestrictive assumption that the

tubes do not possess initial material torsion, the frame,

is obtained by sliding

(0)

without rotation about its z axis (i.e., a Bishop frame [14]).

As the tubes move, bend and twist, these material frames act

as body frames tracking the displacements of their cross

sections. It is also useful to define a reference frame,

which displaces with the cross sections, but does not rotate

about its z axis under tube torsion.

As the

centerline, it experiences a body-frame angular rate of

change per unit arc length given by

é

ëê

in which (,)

ixiy

uu

are the components of curvature due to

bending and

iz

u is the curvature component due to torsion. A

circumflex on a curvature component is used to designate

the initial pre-curvature of a tube.

The kinematic input variables consist of the rotation and

translation of each tube about and along the common

centerline of the combined tubes. The rotation angle, qi(s) ,

is defined as the z -axis rotation angle from frame

frame

( )

iF

( )

iF s ,

iF

along the tube centerline

0( )F s ,

thi tube’s coordinate frame

( )

iF s slides down its

ui(s)=

uix(s)uiy(s)uiz(s)

ù

ûú

T (1)

0( )F s to

iF s . The translation variable, li, is defined as the

arc length distance from frame F0(0) to the initially

coincident frame Fi(0) . In the rest of the paper, all vector

quantities associated with tube i , e.g.,

with respect to frame

( )

robot, e.g., net bending moment, are written with respect to

frame

F s .

As shown in the figure, insertion of one tube inside the

other causes each to bend and twist along their length. The

application of externally applied wrenches generates

additional bending and twisting of the tubes.

( )

iu s , are written

iF s . Vectors associated with the

0( )

A. Derivation of Multi-tube Model

The quasistatic model including external loading can be

derived by combining three equations – a constitutive model

relating bending moments to changes in curvature of

individual tubes, the equilibrium of bending moments and

shear forces on the cross section of the assembled tubes, and

a compatibility equation relating the individual curvatures of

the assembled tubes. Additional equations are needed to

compute the net shear force and bending moment on the

robot as a function of arc length.

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The constitutive model and compatibility equations are

independent of the external loading and so are identical to

those of the unloaded kinematic model presented in [8]. The

equilibrium equation of [8], however, must be modified to

include the net bending moment arising from external loads.

Furthermore, to compute net bending moment, new

differential equations must be introduced to compute both it

and net shear force. Each is described below.

Fig. 2. Tube coordinate frames are denoted

( )

iF s . The relative z-axis twist

( )( )

.

angle between tube frame

(1) Constitutive Model: When a tube with initial

curvature ˆ ( )

bending moment is generated. Assuming linear elastic

behavior, the bending moment vector

along tube i is given by

( )

ii

m sK u s

Given the coordinate frame convention described above, all

vectors are expressed with respect to frame

the frame-invariant stiffness tensor given by

00

00

00

iz

k

in which

moment of inertia,

is the shear modulus of tube i.

(2) Compatibility of Deformations: Assuming that the

clearance between each pair of adjacent tubes is just

sufficient to enable relative motion, all tubes must conform

to the same final x-y (bending) curvature. Each tube is free,

however, to twist independently about its z axis. The z

component of curvature,

( )

i

u s

change of twist angle with respect to arc length,

0( )F s and frame

iF s is

is

iu s is deformed to a different curvature

( )

iu s , a

( )

i

m s at any point s

ˆ

( )( )

ii

u s

(2)

( )

iF s , and

i

K is

00

00

00

ixi i

iiyi i

ii

k E I

KkE I

J G

(3)

iE is the modulus of elasticity,

iJ is the polar moment of inertia and

iI is the area

i

G

z

, equates to the rate of

( )

is

,

( )s( )

ii

z

u s

. (4)

The resulting bending curvatures can be equated when

written in the same frame. Expressing tube curvatures in

terms of the robot frame curvature,

in which

( )(3)

zi

RSO

is a rotation about the z axis by

angle

[0,0,1]T

.

(3) Equilibrium of Bending Moments: On each cross section,

the bending moments in each tube must sum to the robot’s

net bending moment,

m s , generated by the external

loading.

n

m sR

As in (5),

( )

zi

R is used to transform tube bending moments

from frame

( )F s .

Combining (2) and (6) expresses net bending moment in

terms of tube curvatures,

n

m sRs K s u s

Solving (5) and (7) for

u s provides an expression for

robot curvature in terms of initial tube curvatures and net

bending moment,

1

( )( )

i

i

Since frame

F s by definition does not rotate about its

z axis,

0

0

z

u

, and so this equation can written in two

parts as

0u , results in

( )

i

s e

0

( )( )( )

T

ziiz

u sRu s

(5)

i and

ze

0( )

0

1

( )( )

( )

zii

i

m s

(6)

iF s to frame

0( )

0

1

ˆ

( )( ( ))

i

( )( ( )( ))

ziii

i

u s

. (7)

0( )

0

1

0

1

ˆ

( ( ))

i

( ) ( )( )

s K s e

( )( )

n

n

ziiiiz

i

u sK s

Rs K s u sm s

(8)

0( )

1

00

,

11

,

ˆ

( )( )( ( ))

i

( ) ( )( )

nn

izii

x y

ii

x y

u sK sRs K s u sm s

(9)

0

11

( )( ) ( )s

( )s u s( )

nn

iziiziz

z

ii

m sksk

(10)

Equations (5) and (9) enable the computation of the x and y

curvatures of all tubes using

,

( )

iz

x y

u sR

An expression is also needed to compute the z curvature of

all tubes,

, 1,,

izi

uin

. Such an expression can be

obtained from the equilibrium equation of the special

Cosserat rod model [15]-[17]. Setting time dependent terms

to zero, the body-frame equilibrium equations for a curved

rod undergoing distributed loading of t Î3torque per unit

length and f Î 3force per unit length can be applied to

each tube

( )( )[ ( )] [ ( )]

( )( )

ii

n sf s

0

,

( ) ( )

i

u s

T

x y

(11)

( )

( )0[ ( )]

i

u s

iiiii

i

m s

su sv sm s

n s

(12)

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Derivatives are with respect to arc length along the rod, s,

and

,

ii

m n Î are the bending moment and shear force

vectors acting on the tube’s cross section. Here, and in the

remainder of the paper, the square brackets on the vectors

iu and

0

[ ]

i iz

uu

u

Consistent with the previous notation,

the angular and linear strain rates per unit arc length,

respectively, experienced by the tube's cross section. Thus,

as described previously,

( )

Similarly, the x and y components of

strain components of the cross section while the z

component is

1

iziz

v

strain. Given the assumptions of negligible shear and

longitudinal strain,

( )0

It can be helpful to note that

to body-frame angular and linear velocities if time is

substituted for arc length. Wrenches applied at either end of

the rod enter the equations as boundary conditions.

Since tube interaction is limited to distributed forces,

( )0

t= in (12) and, for each tube, it reduces to

[ ]

ii

m u m

To eliminate moments from these equations, we can use the

constitutive model for moments (2) and its derivative to

arrive at

ˆ

( ) ( )( ( )/

izizz

usus k s k s

k sk su

3

iv denote the skew-symmetric form

0

0

iz iy

u

ix

iy ix

uu

u

(13)

3

( ), ( )u s v s Î are

ii

iu s has the units of curvature.

( )

iv s are the shear

in which

iz

is the longitudinal

0 1

T

iv s

(14)

( )

iu s and

( )

iv s are analogous

is

[ ]v n

iii

(15)

ˆ

u s

( ))(( )( ))

ˆˆ

(( )/( )) ( )

s u

( )

s

( )

s u

( )

s

zzz

xzixiyiyix

u s

u

(16)

Equations (4) and (16) are a set of second order differential

equations for the tubes’ twist angles,

integrated using the algebraic equations (9) and (11). These

equations are identical to those describing the unloaded

kinematic model except that (9) now includes the net

bending moment on the tubes,

(4) Net Bending Moment and Shear Force: While (11)

provides the z component of

components that are needed for (9). To compute net bending

moment as a function of arc length, the equilibrium special

Cosserat model (12) can be applied again, but this time to

the collection of tubes.

( )( )[ ( )] [ ( )]

( )( )n s f s

Since net bending moment on the robot’s cross section

evolves together with net shear force,

simultaneously integrated. Here

externally applied distributed torque and force per unit

length of the robot as shown in Fig. 3.

i , that must be

0( )m s [8].

0( )m s , it is the x and y

00000

0000

( )

( )0[ ( )]u s

m s

su sv sm s

n s

(17)

0( )n s , both must be

and

f s are the

0( ) s

0( )

Fig. 3. External loading on robot consists of distributed forces,

0( )f s , and

distributed moments,

0( ) s

t

, as well as concentrated forces,

0( )n L , and

concentrated moments,

Robot curvature,

applies to all tubes comprising the robot,

Equations (4),(9)-(11),(16)-(18) form a set of equations in

the state variables

( ),

ij

s

2,,jn

. Observe that

1

( )

algebraically from (10).

The boundary conditions for the state variables are split

between the proximal and distal ends of the robot.

(0) actuator positions

i

LuL

m L

n L

The x and y components of

( )

and (11). While (10) evaluated at s

expression for the weighted sum of

insufficient to solve for the individual values of

can be resolved by assuming that the total external twisting

moment is applied to a single tube, say tube j. The resulting

values for

( )

iz

uL are given by

0( )/

( )

0

Physically, this situation corresponds to tube j extending

slightly beyond the other tubes so that it comprises the tip of

the robot.

There is, in fact, no reason that the tubes must be of the

same length. The equations above apply to any telescoping

arrangement of tubes in which the stiffness and pre-

curvature of each tube can be an arbitrary function of arc

length. This includes discontinuities in both stiffness and

pre-curvature. Consequently, there is no need to subdivide

the domain during integration over a telescoping

arrangement of tubes. Distal to the physical end of each

tube, its stiffness and curvature can be defined as zero.

0( )m L .

0( )u s , is defined by (8) and since (14)

0( ) v s

00 1

T

. (18)

00

( ),s m s n s( ), ( )

,

1,,in

;

1

( )s

z

su

can be computed

0

0

( )

( )

( )

( )

body frame external tip moment

body frame external tip force

i iz

(19)

iu L can be computed from (9)

L

( ),

iz

u L i

provides an

1,,n

( )

iz

uL . This

, it is

,

ziz

iz

mLki

i

j

j

u L

. (20)

B. Numerical Solution of Multi-tube Model

When solving the multi-tube equations given by (4),(9)-(11),

(16)-(18) together with boundary conditions defined by (19)

and (20), three issues must be considered. First, the

boundary conditions are split between the ends of the tubes.

Second, integration of (

0

( )v s

)

3

0

, ( )(3)u s so

ÎÎ

to obtain

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the robot coordinate frame,

that it evolves on

in these equations is expressed in the body coordinate frame,

it is often convenient to express external loads with respect

to a different frame. Each of these issues is addressed in the

paragraphs below.

(1) Split Boundary Conditions: The problem of split

boundary conditions is one that has been addressed with the

unloaded kinematic equations. In fact, the unloaded

equations can be recovered by setting the external loading to

zero [8].

00

( )( )s f sm L

t==

While such equations can be solved by a variety of standard

means, one approach is to pose the forward kinematics as a

root finding problem in which guesses of

integrate from

0

sL

until the desire values of

obtained.

(2) Integration on SE(3): Integrating the unloaded

kinematics required integrating tube curvatures with respect

to arc length. Analogous to integrating body frame twist

velocity, numerical integration of ui and vimust preserve

the group structure of SE(3). A variety of numerical

integration methods are available for this purpose

[16],[18],[19].

(3) External-Load Coordinate Frame: It is often desirable

to express the loading in different coordinates than the body

frame coordinates of (19) and (20). In this case, however,

the boundary condition is a function of the shape of the

robot. For example, suppose it is desired to produce a tip

wrench that is specified with respect to the base frame of the

robot, F0(0) . Then the body-frame tip wrench, written with

respect to frame F0(L) is related to the desired world frame

tip wrench, written with respect to frame F0(0) , by

( )

0

0

( )

( )

[

L

m L

Rp

in which R0Land

frame F0(L) with respect to frame F0(0) . In this case, the

equations must be solved iteratively with respect to both tip

wrench and actuator positions.

0( )F s , must be performed such

. Thirdly, while the external loading

(3)SE

00

( )( )0 n L

== (21)

( )

iL

are used to

(0)

i

are

00

(0)

0

00

000

( )

( )

0

]

F LF

T

L

TT

LL

n Ln L

m L

R

R

(22)

0L

p describe the orientation and position of

III. APPROXIMATE SINGLE-TUBE MODEL

In contrast to the model presented above, references [11]-

[13] propose an approach in which the load-deflected shape

of a continuum robot is computed as the sequence of two

transformations. The first employs an unloaded kinematic

model to compute the

configuration together with the external loading are the

inputs to a second transformation that computes the

deflected shape by modeling the robot as a single rod with

its stiffness given by the composite stiffness of the robot’s

elements. While approximate since it ignores internal

displacements arising from loading, its solution takes the

form of an initial value problem and so can be computed

robot configuration. This

efficiently.

The equations to be solved are a subset of those for the

multi-tube model and consist of (2), (17), (18) which are

repeated here for clarity.

( ) ( )[

( )( )n s f s

00

( )( )u sKs m s

0v s

As before,

m s and

n s are the net bending moment and

shear force on the robot as functions of arc length. Robot

curvature,

u s , described in coordinate frame

algebraically related to

m s . The composite robot

stiffness,

K s , is defined as the effective bending and

torsional stiffness of the robot cross section as a function of

arc length.

The initial robot curvature,

output of the unloaded forward kinematic model. The

boundary conditions for these equations are given by a

subset of (19) consisting of the applied tip force and bending

moment.

( ) body frame external tip moment

( ) body frame external tip forcen L

Since the boundary conditions are all defined at the distal

end of the robot, they can be solved as an initial value

problem by integrating from the tip back to the base.

The solution is, however, subject to the conditions

described in sections II.B.2 and II.B.3 above. Namely, the

equations must be integrated on

tip loading is not defined with respect to the body frame

then the initial value problem must be solved iteratively to

account for the rotation of the body tip frame in response to

deflection. In real-time use, the number of iterations can be

minimized by using the tip frame rotation from the

preceding time step as the initial guess.

00000

0000

( )] [ ( )]

0[

( )

( )( )]

m s

su sv s

u s

m s

n s

(23)

1

00ˆ

u s ( )( )

(24)

0( )0 1

T

(25)

0( )

0( )

0( )

0( )

F s , is

0( )

0( )

0ˆ ( )

u s , is obtained as the

0

0

m L

(26)

(3)SE

. Furthermore, if the

A. Comparison with Multi-tube Model

The computational costs of the models can be assessed by

considering the total number of state variables and the

locations of the boundary conditions as summarized in Table

1. While not included in the table, it is also necessary for

both models to simultaneously integrate (

obtain the coordinate frame

F s .

Exclusive of

F s , the total number of state variables is

2n+5 for the multi-tube model. Since each tube of a robot

contributes two degrees of freedom corresponding to its

rotation and translation, it requires three tubes to produce a

robot with six degrees of freedom. For such a robot, solution

of the multi-tube model involves integrating eleven state

variables with respect to arc length using split boundary

conditions. The single-tube model possesses six states

regardless of the number of tubes and all boundary

conditions are at the distal end.

Since computation of the unloaded kinematic model

)

00

( ),( )v s u s

to

0( )

0( )

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